Borel's Law and the Origin of Many Creationist Probability Assertions

Creationist Karl Crawford presents a probability argument against the random
formation of life that makes reference to Borel's Law.

ften on talk.origins we have seen assertions to the
effect that there exists a law that is well known to physicists
and/or mathematicians (possibly implying that it is a mathematical
theorem) that there is a particular order of probability below
which any event is considered to be "essentially impossible". This
statement usually preceeds a calculation based on some unrealistic
model of the formation of complicated organic molecules via the
random assembly of atoms as "proof" that abiogenesis is impossible.
At the end of this article, references are given to several creationist sources
that refer to this probability assertion as "Borel's Law".

Conclusions of this FAQ

The "law" in question does not exist as a mathematical theorem, nor
is there a universally decided upon "minimum probability" among
the physical sciences community. Rather, Borel's Law originated
in a discussion in a book written by Emil Borel for non-scientists.
Borel shows examples of the kind of logic that any scientist might use to
generate estimates of the minimum probability below which events
of a particular type are considered negligible.
It is important to stress that each of these estimates are created
for specific physical problems, not as a universal
law.

A Discussion of Karl Crawford's Original Post

A post by t.o regular, creationist Karl Crawford (aka ksjj),
shed some light on the possible origin of this "law".

Talk.origins regulars, of course, will recognize that all of the models that are
used to generate the tremendously tiny odds are based on faulty assumptions.
However, the point in question is the reference to mathematician Emil Borel:

...Mathematicians generally agree that, statistically, any odds beyond 1 in
1050 have a zero probability of ever happening.... This is Borel's law in action which was derived by
mathematician Emil Borel....

I was intrigued by the reference to Emil Borel. While Borel is famous
in mathematical circles, he is hardly a household name, so I
wanted to see if there was such a thing as "Borel's law" in the
subject of probability and statistics. After searching a number of
probability and statistics textbooks, technical treatises, and other
scholarly works on the subject without finding any reference to such
a thing, I happened quite by chance (no pun intended) on two books by Borel,
himself.

A Discussion of Borel's Law

The first is Probability and Life, a 1962 Dover English translation
of the French version published in 1943 as Le Probabilites et la Vie.
The second is Probability and Certainty, a 1963 Dover English
translation of the French version published in 1950 as
Probabilite et Certitude. Both of these books are
"science for the non-scientist" type books rather than scholarly
treatments of the theory of probability.

In Probability and Life, Borel states a "single law of chance"
as the principle that "Phenomena with very small probabilities do not occur".
At the beginning of Chapter Three of this book, he states:

When we stated the single law of chance, "events whose probability
is sufficiently small never occur," we did not conceal the lack of
precision of the statement. There are cases where no doubt is
possible; such is that of the complete works of Goethe being reproduced
by a typist who does not know German and is typing at random. Between
this somewhat extreme case and ones in which the probabilities are very
small but nevertheless such that the occurrence of the corresponding
event is not incredible, there are many intermediate cases. We shall
attempt to determine as precisely as possible which values of probability
must be regarded as negligible under certain circumstances.

It is evident that the requirements with respect to the degree of
certainty imposed on the single law of chance will vary depending
on whether we deal with scientific certainty or with the certainty
which suffices in a given circumstance of everyday life.

The point being, that Borel's Law is a "rule of thumb" that exists on
a sliding scale, depending on the phenomenon in question. It is
not a mathematical theorem, nor is there any hard number that draws a
line in the statistical sand saying that all events of a given probability
and smaller are impossible for all types of events.

Borel continues by giving examples of how to choose such cutoff probabilities.
For example, by reasoning from the traffic death rate of 1 per million
in Paris (pre-World War II statistics) that an event of probability of 10-6
(one in a million) is negligible on a "human scale". Multiplying this
by 10-9 (1 over the population of the world in the 1940s), he obtains
10-15 as an estimate of negligible probabilities on a "terrestrial
scale".

To evaluate the chance that physical laws such as Newtonian mechanics
or laws related to the propagation of light could be wrong, Borel
discusses probabilities that are negligible on a "cosmic scale", Borel
asserts that 10-50 represents a negligible event on the cosmic
scale as it is well below one over the product of the number of observable
stars (109) times the number of observations that humans could make
on those stars (1020).

To compute the odds against a container containing a mixture of
oxygen and nitrogen spontaneously segregating into pure nitrogen
on the top half and pure oxygen on the bottom half, Borel states
that for equal volumes of oxygen and nitrogen the odds would be
2-n where n is the number of atoms, which Borel states as being
smaller than the negligible probability of 10-(10(-10)),
which he assigns as the negligible probability on a "supercosmic"
scale. Borel creates this supercosmos by nesting our universe U1
inside successive supercosmoses, each with the same number of elements identical to
the preceding cosmos as that cosmos has its own elements, so
that U2 would be composed of the same number of U1's as U1 has atoms,
and U3 would be composed of the same number of U2's as U2 has U1's,
and so forth on up to UN where N=1 million. He then creates a similar
nested time scale with the base time of our universe being a billion
years (T2 would contain a billion, billion years) on up to TN, N=1 million.
Under such conditions of the number of atoms and the amount of time,
the probability of separating the nitrogen and oxygen by a random process
is still so small as to be negligible.

Ultimately, the point is that the user must design his or her "negligible
probability" estimate based on a given set of assumed conditions.

Curiously, in spite of the suggestive title of the book Probability and Life, Borel has
no discussion of evolution or abiogenesis-related issues. However,
in Probability and Certainty, the last section of the main text
is devoted to this question.

From Probability and Certainty, p. 124-126:

The Problem of Life.

In conclusion, I feel it is necessary to say a few words
regarding a question that does not really come within
the scope of this book, but that certain readers might nevertheless
reproach me for having entirely neglected. I mean the problem of
the appearance of life on our planet (and eventually on other planets
in the universe) and the probability that this appearance may have
been due to chance. If this problem seems to me to lie outside
our subject, this is because the probability in question is too
complex for us to be able to calculate its order of magnitude. It
is on this point that I wish to make several explanatory comments.

When we calculated the probability of reproducing by mere chance
a work of literature, in one or more volumes, we certainly observed
that, if this work was printed, it must have emanated from a human
brain. Now the complexity of that brain must therefore have been
even richer than the particular work to which it gave birth. Is it
not possible to infer that the probability that this brain may have
been produced by the blind forces of chance is even slighter than
the probability of the typewriting miracle?

It is obviously the same as if we asked ourselves whether we could
know if it was possible actually to create a human being by combining
at random a certain number of simple bodies. But this is not
the way that the problem of the origin of life presents itself: it
is generally held that living beings are the result of a slow process
of evolution, beginning with elementary organisms, and that this
process of evolution involves certain properties of living matter that
prevent us from asserting that the process was accomplished in
accordance with the laws of chance.

Moreover, certain of these properties of living matter also belong
to inanimate matter, when it takes certain forms, such as that of
crystals. It does not seem possible to apply the laws of probability
calculus to the phenomenon of the formation of a crystal in a
more or less supersaturated solution. At least, it would not be
possible to treat this as a problem of probability without taking
account of certain properties of matter, properties that facilitate
the formation of crystals and that we are certainly obliged to verify.
We ought, it seems to me, to consider it likely that the formation
of elementary living organisms, and the evolution of those organisms,
are also governed by elementary properties of matter that we do
not understand perfectly but whose existence we ought nevertheless
admit.

Similar observations could be made regarding possible attempts to
apply the probability calculus to cosmogonical problems. In this
field, too, it does not seem that the conclusions we have could
really be of great assistance.

In short, Borel says what many a talk.origins poster has said
time and time again when confronted with such creationist arguments:
namely, that probability estimates that ignore the non-random elements predetermined
by physics and chemistry are meaningless.